The Choice of the Null and Alternative Hypotheses: What is the Claim?

One problem faced by students when solving a Stats test or working on a homework is to identify the null and alternative hypotheses and the state the claim, based on the wording of a specific situation described on a problem that is posed to you. This type of question can be tricky, especially when you don't know exactly what they mean by "the claim".

The Null and Alternative Hypothesis ARE claims

Let us clarify a few things: First, the two ultimate elements of a hypothesis test are the null hypothesis and the alternative hypotheses. Those two are CLAIMS, in the English language sense, because they both make claims about a population parameter. For example, say that we have \({{H}_{0}}:\mu =10\) and \({{H}_{A}}:\mu \ne 10\). In that case we see that the null hypothesis \({{H}_{0}}\) is "claiming" that the population mean \(\mu\) is equal to 10, whereas the alternative hypothesis \({{H}_{A}}\) is "claiming" that the population mean \(\mu\) is different than 10.

So, then, if both the null and alternative hypotheses are claims, what is the claim you are being asked about then? Well, the answer is depends on the setting of the problem, but there is something we know: the claim that is referred in a hypothesis test question will be either the null OR the alternative hypothesis, and it based on what theresearcher wants to prove.

Summarizing

Let us repeat: The claim that you are being asked about will be either the null hypothesis OR the alternative hypothesis (one or the other), and it will correspond to what needs to be proven, or what the researcher claims to be true, in the context of the problem.

That's it. Super simple, huh?

There is a little technical issue, though, that is worth mentioning. In the context of a hypothesis testing, we cannot "prove" the null hypothesis. The idea of hypothesis testing is to assess whether or not sample data departs significantly with what would be expected if the null hypothesis were true. So all that the sample evidence can do is to assess whether, based on the evidence collected, the null hypothesis seems to be contradicted by it. If that is the case, then we reject the null hypothesis. If not, we do not "accept the null hypothesis", we simply FAIL TO REJECT IT.

EXAMPLE

Be careful when a question says, for example: "a hypothesis test was conducted to assess whether the population mean is equal to 10". In that case, we would have that the null hypothesis is \({{H}_{0}}:\mu =10\) and the alternative hypothesis is \({{H}_{A}}:\mu \ne 10\). What would be the claim? The question says that what needs to be proven is whether or not the population mean is equal to 10. So, then, the claim in this case corresponds to the null hypothesis \({{H}_{0}}\). Also, notice that in this case, the hypothesis test cannot "prove" that \(\mu =10\). The only thing you can do is to check sample data to see if you enough evidence to claim that the sample is too unlikely to have happened if the null hypothesis were true, in which case, you reject the null hypothesis, and in the context of the example, you reject the claim that the population mean is true. If the null hypothesis not rejected, you will not say that you support the null hypothesis. All you can say is that "you fail to reject the null hypothesis". In this example the claim was actually the null hypothesis, so then you would say "I don't have enough evidence to support the claim".